Subgroups — C₁₁₃×#10727 #43278747

Order 28928.

1678 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₁₁₃	C₂	1	2	yes	maximal
3	C₂×#343×C₁₁₃	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₁₃	C₄	1	2	yes	—
5	C₂⁴×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₁₃	C₂²	1	4	yes	—
7	C₂²×C₄⋊C₄×C₁₁₃	—	2	4	no	—
8	C₂²×C₄⋊C₄×C₁₁₃	—	2	8	no	—
9	C₂²×C₄⋊C₄×C₁₁₃	C₄	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
11	#407×C₁₁₃	—	2	4	no	—
12	#407×C₁₁₃	C₄	1	8	yes	complemented
13	C₂×C₂²⋊C₈×C₁₁₃	—	2	4	no	—
14	#343×C₁₁₃	—	2	8	no	—
15	C₂⁵×C₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₁₃	—	2	4	no	—
17	C₂³×C₄×C₁₁₃	—	2	4	no	—
18	C₂³×C₄×C₁₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₁₃	—	2	8	no	—
20	C₂³×C₄×C₁₁₃	—	4	8	no	—
21	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
25	C₂³×C₄×C₁₁₃	D₄	1	4	yes	—
26	C₂³×C₄×C₁₁₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₁₃	—	2	4	no	—
29	C₂×M₁₆×C₁₁₃	—	2	8	no	—
30	C₂×M₁₆×C₁₁₃	—	2	8	no	—
31	C₂×M₁₆×C₁₁₃	C₂×C₄	1	8	yes	complemented
32	C₂²×C₈×C₁₁₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	32	no	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₁₃	D₄	1	8	yes	—
39	C₂²⋊C₈×C₁₁₃	—	2	16	no	—
40	C₂⁴×C₁₁₃	—	2	4	no	—
41	C₂⁴×C₁₁₃	—	2	4	no	—
42	C₂⁴×C₁₁₃	—	2	2	no	—
43	C₂⁴×C₁₁₃	—	2	2	no	—
44	C₂⁴×C₁₁₃	C₄²	1	2	yes	—
45	C₂⁴×C₁₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂⁴×C₁₁₃	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₁₃	—	2	16	no	—
51	C₂²×C₄×C₁₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₁₃	—	2	16	no	—
56	C₂²×C₄×C₁₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₁₃	—	4	32	no	—
59	C₂²×C₄×C₁₁₃	—	4	8	no	—
60	C₂²×C₄×C₁₁₃	—	4	8	no	—
61	C₂²×C₄×C₁₁₃	—	4	8	no	—
62	C₂²×C₄×C₁₁₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
66	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
68	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
74	M₁₆×C₁₁₃	—	2	16	no	—
75	M₁₆×C₁₁₃	—	4	8	no	—
76	C₂×C₈×C₁₁₃	—	2	8	no	—
77	C₂×C₈×C₁₁₃	—	4	4	no	—
78	C₂×C₈×C₁₁₃	—	4	4	no	—
79	C₄⋊C₄×C₁₁₃	—	2	16	no	—
80	C₄⋊C₄×C₁₁₃	—	2	16	no	—
81	C₂³×C₁₁₃	—	2	8	no	—
82	C₂³×C₁₁₃	—	2	16	no	—
83	C₂³×C₁₁₃	—	2	8	no	—
84	C₂³×C₁₁₃	—	2	8	no	—
85	C₂³×C₁₁₃	—	2	8	no	—
86	C₂³×C₁₁₃	—	2	4	no	—
87	C₂³×C₁₁₃	—	2	2	no	—
88	C₂³×C₁₁₃	—	2	4	no	—
89	C₂³×C₁₁₃	—	2	2	no	—
90	C₂³×C₁₁₃	—	2	2	no	—
91	C₂³×C₁₁₃	—	2	2	no	—
92	C₂³×C₁₁₃	—	2	2	no	—
93	C₂³×C₁₁₃	—	2	2	no	—
94	C₂³×C₁₁₃	#95	1	4	yes	—
95	C₂³×C₁₁₃	#100	1	4	yes	—
96	C₂³×C₁₁₃	#101	1	4	yes	—
97	C₂³×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₁₃	#117	1	2	yes	—
102	C₂×C₄×C₁₁₃	—	2	4	no	—
103	C₂×C₄×C₁₁₃	—	2	2	no	—
104	C₂×C₄×C₁₁₃	—	2	4	no	—
105	C₂×C₄×C₁₁₃	—	2	2	no	—
106	C₂×C₄×C₁₁₃	—	2	16	no	—
107	C₂×C₄×C₁₁₃	—	2	16	no	—
108	C₂×C₄×C₁₁₃	—	2	8	no	—
109	C₂×C₄×C₁₁₃	—	4	32	no	—
110	C₂×C₄×C₁₁₃	—	4	8	no	—
111	C₂×C₄×C₁₁₃	—	4	8	no	—
112	C₂×C₄×C₁₁₃	—	4	8	no	—
113	C₂×C₄×C₁₁₃	#95	1	4	yes	—
114	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₁₃	C₄×D₄	1	8	yes	—
117	C₂×C₄×C₁₁₃	#120	1	4	yes	—
118	C₂×C₄×C₁₁₃	#121	1	4	yes	—
119	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₁₁₃	#123	1	4	yes	—
121	C₈×C₁₁₃	—	4	8	no	—
122	C₂²×C₁₁₃	—	2	16	no	—
123	C₂²×C₁₁₃	—	2	16	no	—
124	C₂²×C₁₁₃	—	2	4	no	—
125	C₂²×C₁₁₃	—	2	4	no	—
126	C₂²×C₁₁₃	—	2	4	no	—
127	C₂²×C₁₁₃	—	2	4	no	—
128	C₂²×C₁₁₃	—	2	4	no	—
129	C₂²×C₁₁₃	—	2	2	no	—
130	C₂²×C₁₁₃	—	2	2	no	—
131	C₂²×C₁₁₃	—	2	4	no	—
132	C₂²×C₁₁₃	—	2	2	no	—
133	C₂²×C₁₁₃	—	2	4	no	—
134	C₂²×C₁₁₃	—	2	2	no	—
135	C₂²×C₁₁₃	#343	1	8	yes	complemented
136	C₂²×C₁₁₃	C₂×#95	1	1	yes	characteristic
137	C₂²×C₁₁₃	C₄×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₁₁₃	#379	1	1	yes	characteristic
139	C₂²×C₁₁₃	#380	1	2	yes	—
140	C₂²×C₁₁₃	#381	1	1	yes	characteristic
141	C₂²×C₁₁₃	C₂×#100	1	2	yes	—
142	C₂²×C₁₁₃	C₂×#101	1	2	yes	—
143	C₄×C₁₁₃	—	2	4	no	—
144	C₄×C₁₁₃	—	4	8	no	—
145	C₄×C₁₁₃	#385	1	4	yes	—
146	C₄×C₁₁₃	#386	1	4	yes	—
147	#10727	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₁₁₃	—	2	8	no	—
149	C₂×C₁₁₃	—	2	2	no	—
150	C₂×C₁₁₃	—	2	2	no	—
151	C₂×C₁₁₃	C₂²⋊#95	1	1	yes	characteristic
152	C₂×C₁₁₃	C₂×#343	1	2	yes	—
153	C₂×C₁₁₃	#1783	1	1	yes	characteristic
154	C₂×C₁₁₃	#1784	1	1	yes	characteristic
155	C₂×C₁₁₃	#1821	1	2	yes	—
156	C₂³×C₄⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic
157	C₂×#407	C₂×C₁₁₃	1	2	yes	—
158	C₂×#343	C₂×C₁₁₃	1	4	yes	complemented
159	C₁₁₃	#10727	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴×C₄	C₄×C₁₁₃	1	2	yes	—
161	C₂⁴×C₄	C₂²×C₁₁₃	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₁₁₃	1	4	yes	—
163	C₂²×C₄⋊C₄	—	2	4	no	—
164	C₂²×C₄⋊C₄	—	2	8	no	—
165	C₂²×C₄⋊C₄	C₄×C₁₁₃	1	2	yes	—
166	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
167	#407	—	2	4	no	—
168	#407	C₄×C₁₁₃	1	8	yes	complemented
169	C₂×C₂²⋊C₈	—	2	4	no	—
170	#343	—	2	8	no	—
171	C₂⁵	C₂×C₄×C₁₁₃	1	1	yes	characteristic
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	4	8	no	—
177	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
178	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
179	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
180	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
181	C₂³×C₄	D₄×C₁₁₃	1	4	yes	—
182	C₂³×C₄	Q₈×C₁₁₃	1	2	yes	—
183	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	8	no	—
186	C₂×M₁₆	—	2	8	no	—
187	C₂×M₁₆	C₂×C₄×C₁₁₃	1	8	yes	complemented
188	C₂²×C₈	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	32	no	—
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	D₄×C₁₁₃	1	8	yes	—
195	C₂²⋊C₈	—	2	16	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	2	no	—
199	C₂⁴	—	2	2	no	—
200	C₂⁴	C₄²×C₁₁₃	1	2	yes	—
201	C₂⁴	C₂²⋊C₄×C₁₁₃	1	2	yes	—
202	C₂⁴	C₄⋊C₄×C₁₁₃	1	2	yes	—
203	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	32	no	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	—	4	8	no	—
217	C₂²×C₄	—	4	8	no	—
218	C₂²×C₄	C₄²×C₁₁₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
221	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
222	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
228	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
229	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
230	M₁₆	—	2	16	no	—
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	8	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₄⋊C₄	—	2	16	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	2	16	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	#95×C₁₁₃	1	4	yes	—
251	C₂³	#100×C₁₁₃	1	4	yes	—
252	C₂³	#101×C₁₁₃	1	4	yes	—
253	C₂³	C₂×C₄²×C₁₁₃	1	1	yes	characteristic
254	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
255	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
256	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
257	C₂³	#117×C₁₁₃	1	2	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	32	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	#95×C₁₁₃	1	4	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₁₁₃	1	8	yes	—
273	C₂×C₄	#120×C₁₁₃	1	4	yes	—
274	C₂×C₄	#121×C₁₁₃	1	4	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	4	yes	—
276	C₂×C₄	#123×C₁₁₃	1	4	yes	—
277	C₈	—	4	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	2	16	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	#343×C₁₁₃	1	8	yes	complemented
292	C₂²	C₂×#95×C₁₁₃	1	1	yes	characteristic
293	C₂²	C₄×C₂²⋊C₄×C₁₁₃	1	2	yes	—
294	C₂²	#379×C₁₁₃	1	1	yes	characteristic
295	C₂²	#380×C₁₁₃	1	2	yes	—
296	C₂²	#381×C₁₁₃	1	1	yes	characteristic
297	C₂²	C₂×#100×C₁₁₃	1	2	yes	—
298	C₂²	C₂×#101×C₁₁₃	1	2	yes	—
299	C₄	—	2	4	no	—
300	C₄	—	4	8	no	—
301	C₄	#385×C₁₁₃	1	4	yes	—
302	C₄	#386×C₁₁₃	1	4	yes	—
303	C₂	—	2	8	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	C₁₁₃×C₂²⋊#95	1	1	yes	characteristic
307	C₂	C₂×#343×C₁₁₃	1	2	yes	—
308	C₂	C₁₁₃×#1783	1	1	yes	characteristic
309	C₂	C₁₁₃×#1784	1	1	yes	characteristic
310	C₂	C₁₁₃×#1821	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₁₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁₃
prime radical	C₂⁵×C₁₁₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂⁴×C₄×C₁₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#343×C₁₁₃	C₂	semidirect
12. #407×C₁₁₃	C₄	semidirect
31. C₂×M₁₆×C₁₁₃	C₂×C₄	semidirect
135. C₂²×C₁₁₃	#343	semidirect
147. #10727	C₁₁₃	direct (N × H)
158. C₂×#343	C₂×C₁₁₃	semidirect
159. C₁₁₃	#10727	direct (N × H)
168. #407	C₄×C₁₁₃	semidirect
187. C₂×M₁₆	C₂×C₄×C₁₁₃	semidirect
291. C₂²	#343×C₁₁₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄⋊C₄×C₁₁₃	2
C₂×#407×C₁₁₃	2
C₂×#343×C₁₁₃	2
#10727	113

Subgroups — C113×#10727 #43278747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#10727 #43278747